Given a weak \(T\) -linkage \(\mathcal {W}\)  of \(G\)  satisfying the properties stated in Lemma REF ,
we first compute the flow function \(\phi \)  representing \(\mathcal {W}\) . Then
we let \(\Pi \)  be the set of all face-edge paths each consisting of faces and edges sharing a common vertex,
and let \(\Gamma : \Pi \rightarrow \Sigma ^*\)  be the function with \(\Gamma (\pi )=\Sigma \cup \Sigma ^{-1}\)  for all \(\pi \in \Pi \) .
By applying the algorithm in [1]}, we can compute a flow function \(\psi \) 
homologous to \(\phi \)  with \(\psi (\pi )\in \Gamma (\pi )\) .
By the argument of [1]}, \(\psi \)  represents a unique \(T\) -linkage \(\mathcal {P}\)  of \(G\) , and thus we can obtain
\(\mathcal {P}\)  from \(\psi \)  in linear time.
